IISC Mathematics Integrated PhD paper 2007

You may download a copy of question paper of IISC Mathematics Integrated PhD paper 2007 held on 29th apr 2007 here ( link no longer exists!)

Finally, an easy one [S]

I caught this good one while I was having a leisurely look at mayhem taunt problems (M74).May look daunting at the first but you should be able to hit it within 10 minutes.

Problem: A circle has 12 equally spaced points on its circumference. How many ways can the numbers 1 through 12 be assigned to the points so that if the points 1 through 12 are connected with line segments, in order, the segments do not cross ? an bad example is here.[S]

An Intriguing One

Its mathematician's intuition, you look at a problem and it gives a resonance of an 'intriguing,deep impact' result. One such thing, I came across recently.

Problem: For any two positive real numbers 'a' and 'b' such that a>=b.Find the maximum number of points that can be included within the circle of radius 'a' which are separated by at least a distance 'b'.

generalized problem said, max number of points within a enclosed 'k' dimensional surface where distance between the points is greater than or equal to a given positive real number.

P.S : I am clueless.